Algebraic Groups I. Grothendieck's theorem on tori (based on notes by S. Lichtenstein) 1. Introduction In the early days of the theory of linear algebraic groups, the ground field was assumed to be algebraically closed (as in work of Chevalley). After experts acquired some experience, the needs of number theory and finite group theory (“finite simple groups of Lie type"!) led them to escape this hypothesis, and they were able to get the theory of connected reductive groups off the ground over a perfect field (using Galois-theoretic techniques to pull things down from the algebraically closed case). The needs of number theory over local and global function fields provided motivation to eliminate the perfectness assumption, but it was not at all clear how to do this. Then Grothendieck came along and in Theorem 1.1 of Expos´eXIV of SGA3 he proved the decisive result which made it possible to make the theory of reductive groups work over an arbitrary field. The result was this: Theorem 1.1 (Grothendieck). Let G be a smooth connected affine group over a field k. Then G contains a maximal k-torus T such that Tk is maximal in Gk. Remark 1.2. The hardest case of the proof is when k is imperfect, and it was for this purpose that Grothendieck's scheme-theoretic ideas in SGA3 were essential, at first. (In Remark 1.5(d) of Exp. XIV, he gave an especially scheme-theoretic second proof for infinite k, invoking the \scheme of maximal tori" which he had constructed earlier and later proved to be rational over k in Theorem 6.1 of Exp. XIV, so he could invoke the elementary fact that rational varieties over infinite fields have rational points!) Borel found the SGA3 proof(s) to be \too technical" for such a concrete result over fields, and in his book he eliminates all the group schemes by using clever Lie-theoretic methods (which amount to working with certain infinitesimal group schemes in disguise, as we shall see). The proof we give is my scheme-theoretic interpretation of the argument in Borel's book [Borel, x18.2(i)]. It is very different from Grothendieck's proof. In class we saw that Theorem 1.1 and torus-centralizer arguments (along with dimension induc- tion) yield the following crucial improvement: Corollary 1.3. For any maximal k-torus T ⊂ G and every field extension K=k, TK ⊂ GK is maximal. In particular, taking K = k, dim T is independent of the maximal k-torus T . The common dimension of the maximal k-tori is called the reductive rank of G because it coincides with the same invariant for the reductive quotient Gk=Ru(Gk). Remark 1.4. We have seen in class that if G is a smooth connected affine k-group such that Gk contains no nontrivial torus then Gk is unipotent (and so by definition G is unipotent). But Corollary 1.3 gives that Gk contains no nontrivial torus if and only if G contains no nontrivial k-torus, so we conclude that G is unipotent if and only if G contains no nontrivial k-torus (the implication \(" being trivial). Beware that if k 6= ks then typically there are many G(k)-conjugacy classes of maximal k-tori, unlike the case of an algebraically closed field. For example, if G = GLn then by HW5 Exercise 5(ii) the maximal k-tori in G are in bijective correspondence with maximal finite ´etalecommutative k-subalgebras of Matn(k). In particular, two maximal k-tori are G(k)-conjugate if and only if the corresponding maximal finite ´etalecommutative k-subalgebras of Mat)n(k) are GLn(k)-conjugate. Hence, if such k-subalgebras are not abstractly k-isomorphic then their corresponding maximal k-tori are not G(k)-conjugate. For example, non-isomorphic degree-n finite separable extension fields of k yield such algebras. Thus, when k 6= ks there are typically many G(k)-conjugacy classes of maximal k-tori in G. 1 2 We saw in class that Gk has no nontrivial tori if and only if Gk is unipotent, so it follows from Grothendieck's theorem that every smooth connected affine k-group is either unipotent or contains a nontrivial k-torus. If all k-tori in G are central then for a maximal k-torus T the quotient G=T is unipotent (as (G=T )k = Gk=Tk contains no nontrivial torus). Hence, in such cases G is solvable. Thus, in the non-solvable case there are always k-tori S whose centralizer ZG(S) (which are always again smooth and connected, by HW8 Exercise 3 for smoothness and discussion in class for connectedness) has lower dimension than G. This enables us to \dig holes" in non-solvable smooth connected k-groups when trying to prove general theorems. Of course, the solvable case has its own bag of tricks (somewhat delicate over imperfect fields). Definition 1.5. For a maximal k-torus T in a smooth connected affine k-group G, the associated Cartan k-subgroup C ⊂ G is C = ZG(T ), the scheme-theoretic centralizer. By the torus-centralizer results from HW8 Exercise 3 and class discussion, Cartan k-subgroups are smooth and connected. Since T is central in its Cartan C, it follows that T is the unique maximal k-torus in C. (Indeed, if there were others then the k-subgroup they generate along with the central T would be a bigger k-torus.) We have C = Z (T ) since the formation of scheme- k Gk k theoretic centralizers commutes with base change, and over k all maximal tori are conjugate. Hence, over k the Cartan subgroups are conjugate, so the dimension of a Cartan k-subgroup is both independent of the choice of Cartan k-subgroup and invariant under extension of the ground field. This number is called the nilpotent rank of G in SGA3, and the rank of G in Borel's book. For example, if G is reductive then it turns out (as is shown in the handout \Unipotent radicals and reductivity") that C = T . That is, in a connected reductive group the Cartan subgroups are precisely the maximal tori. Remark 1.6. It is a very difficult theorem that in any smooth connected affine group G over any field k, all maximal k-split tori are G(k)-conjugate. This is 20.9(ii) in Borel's textbook for reductive G, which we will treat in the sequel course. The general case was announced without proof by Borel and Tits, and is proved as Theorem C.2.3 in the book \ Pseudo-reductive groups". The dimension of a maximal split k-torus is thus also an invariant, sometimes called the k-rank of G (and mainly of interest in the reductive case). As a final comment before we embark on the proof of Theorem 1.1, note that since tori split over a finite separable extension, we have the following important consequence of Theorem 1.1. Corollary 1.7. For a smooth connected affine group G over a field k, there exists a finite Galois 0 0 extension k =k such that Gk0 has a split maximal k -torus. 2. Start of proof of theorem 1.1 We will primary focus on the case in which k is infinite, which ensures that kn ⊂ An is Zariski- k dense, and thus in particular g = Lie(G) is Zariski-dense in gk. The case of finite k requires a completely different argument, using \Lang's theorem", and is explained in Proposition 16.6 of Borel's book. (In general, x16 of Borel's book explains the elegant technique due to Lang which is often useful to overcome difficulties with lack of Zariski-density over finite fields. In SGA3 the case of finite k is likewise handled by using Lang's trick.) We first treat the \easy" case in which Gk has a central maximal torus S. (This case will work over all k, even finite fields.) Since all maximal tori are G(k) conjugate, a unique one is automatically normal in Gk. By HW 6, Exercise 3(ii), a normal torus in a smooth connected k- group is automatically central. (This is basically because the automorphism scheme of a split torus 3 of rank n is the constant group GLn(Z) that is ´etale,so the conjugation action of Gk on a normal torus is classified by a homomorphism to the ´etaleautomorphism scheme, which in turn must be trivial when G is connected. Note the similarity with how one proves the commutativity of the fundamental group of a connected Lie group.) Thus, we are in the situation where there exists a unique maximal k-torus S ⊂ Gk. Our problem is to produce one defined over k. This is rather elementary over perfect fields via Galois descent, but here is a uniform method using group schemes that applies over all fields; this technique will be useful later on as well. 0 Let Z = ZG, the identity component of the scheme-theoretic center of G. Since the formation of the center (and identity component) commutes with base change, we have S ⊂ (Zk)red as a maximal torus in the smooth commutative k-group (Zk)red. By the structure of smooth connected commutative k-groups, it follows that (Zk)red = S × U for a smooth connected unipotent k-group U. For any n not divisible by char(k), consider the torsion subgroup Z[n]. This is a commutative, affine algebraic k-group, and since the derivative of [n]: Z ! Z is n : Lie(Z) ! Lie(Z), it follows that Lie(Z[n]) is killed by n 2 k×.